The spectra of composition operators from linear fractional maps acting upon the Dirichlet space

Let be a non-constant linear fractional transformation (necessarily of the form ). Let denote the Dirichlet space of analytic functions. We determine the spectrum of the composition operator defined by . Eigenfunctions for the operator frequently do not belong to the space . However, spectral results for the operator , much like those that have already been demonstrated for the operator , are presented in this paper.